Trigonometric Inequality: T/F: If 5pi/4 <= x < 3pi/2, then 0 <= cot(x) < 1

[Nexo]

Trigonometric Inequality: T/F: If 5pi/4 <= x < 3pi/2, then 0 <= cot(x) < 1

Hello everyone!
I hope you can help me out with this problem. I've tried everything but the third question doesn't want to be solved. In fact is just one question but there are 4 points within this question and it asks me to show which ones are False (F) and which ones are True (V - from the Spanish word "Verdadero" which means "True"). The answer is the alternative B) and I've been able to solve points I, II and IV. The one that is hard for me is the third one. I'll show you what I've done in each case.
Note: "Si" in Spanish means "If" in English.

I. It's given:

3*pi/8 < x < pi/2 \\ Multiply by 2
3*pi/4 < 2x < pi \\ "Take" Cosine in every member.
Cos(3*pi/4) < Cos2x < Cos(pi/2) \\ Evaluate
-(2^(1/2))/2 < Cos2x < 0 ---> Then I. is False.

II. It's given:

3*pi/2 <= x < 5*pi/2 \\ Divide by 3
pi/2 <= x/3 < 5*pi/6 \\ "Take" Sine in every member.
Sin(pi/2) <= Sin(x/3) < Sin(5*pi/6) \\ Evaluate
1 <= Sin(x/3) < 1/2 <-- Does it make senses? Or whenever I make use of a trigonometric function I should rearrange the values from the smaller to the bigger one, which would result in:
1/2 <= Cos(x/3) < 1 or
1/2 < Cos(x/3) <= 1 ?
The same thing happens in the next point and according to the answer, yes, I should rearrange the values, lets see:

III. It's given:

5*pi/4 < x <= 3*pi/2 \\ "Take" Cotangent in every member.
Cot(5*pi/4) < Cotx <= Cot(3*pi/2) \\ Use Tangent as follows
1/Tan(5*pi/4) < Cotx <= 1/Tan(3*pi/2) \\ Evaluate
1/1 < Cotx <= 1/(-inf)
1 < Cotx <= 0 <-- Again, Does it make senses? Or should I rearrange it which would result in...?
If I leave it like this, this point would be False (F), but if I rearrange, it would be True (V).

IV. It's given:

Cos (5*pi/7) +1 < Sin(15*pi/7)
-0.62 + 1 < 0.43
0.38 < 0.43 ----> This is clearly True (V).

So the only problem I have is with the rearrangement of the interval's limit. Can anyone, please, help me out?

Thanks!

 

[stapel]

...the third question doesn't want to be solved....The answer is the alternative B)...

III. It's given:

5*pi/4 < x <= 3*pi/2 \\ "Take" Cotangent in every member.
Cot(5*pi/4) < Cotx <= Cot(3*pi/2) \\ Use Tangent as follows
1/Tan(5*pi/4) < Cotx <= 1/Tan(3*pi/2) \\ Evaluate
1/1 < Cotx <= 1/(-inf)
1 < Cotx <= 0 <-- Again, Does it make senses? Or should I rearrange it which would result in...?
If I leave it like this, this point would be False (F), but if I rearrange, it would be True (V).

My understanding of your post is that you need to show whether the following is true ("verdad") or false ("falso"):

. . . . .\(\displaystyle \mbox{If }\, \dfrac{5\pi}{4}\, \leq\, x\, <\, \dfrac{3\pi}{2}\, \mbox{ then }\, 0\, \leq\, \cot(x)\, <\, 1\)

You have been given that this is true. You're not sure if you've proven this correctly.

Your first step appears to have been to "take the cotangent of the inequality". However, looking at the graph of cotangent, is it true that the cotangent is increasing on the interval? So is that first step valid?

Instead, try looking at the values of the cotangent at the endpoints, such as they are, and note the direction of the cotangent curve on this interval. Note that there are no asymptotes within this interval, so the curve is connected. Then what must be the values of cotangent on this interval?